One side of the rectangle is 14 cm larger than the other. Find the sides of the rectangle if its diagonal is 26 cm.

1. Let one side of the rectangle be x cm, and the other y cm. Let the side x be greater than the side y. It is known from the problem statement that one of the sides of the rectangle is 14 cm larger than the other. Therefore, we can write

y = x – 14.

2. It is known that the diagonal of a rectangle is 26. From the Pythagorean theorem it follows that the square of the hypotenuse (diagonal) is equal to the sum of the squares of the legs (sides of the rectangle). Therefore, we can write

x ^ 2 + y ^ 2 = 26 ^ 2.

3. Let’s compose and solve the system of equations

y = x – 14,

x ^ 2 + y ^ 2 = 26 ^ 2.

Substituting the first equation into the second, we get

x ^ 2 + (x – 14) ^ 2 = 676,

x ^ 2 + x ^ 2 – 28x + 196 = 676,

2x ^ 2 – 28x – 480 = 0,

x ^ 2 – 14x – 240 = 0.

a = 1, b = -14, c = – 240, k = b / 2 = -7.

D1 = k ^ 2 – ac = 49 + 240 = 289 = 172.

x1 = (- k + √D1) / a = 7 + 17 = 24,

x2 = (- k – √D1) / a = 7 – 17 = -10.

Since the side of the rectangle cannot be negative, we leave the root x = 24. Therefore, y = x – 4 = 24 – 4 = 20.

Answer: the sides of the rectangle are 24 cm and 20 cm.